\chapter{Gröbner Bases}

The theory of Gröbner bases for ideals in polynomial rings was introduced by
Bruno Buchberger \cite{bb}, who named the concept in honor of his advisor
Wolfgang Gröbner (1899--1980). Buchberger also developed the fundamental
algorithm for the computation of a Gröbner basis known as Buchberger's
algorithm. A similar concept for ideals in power series rings was introduced by
Heisuke Hironaka \cite{hh1}, \cite{hh2}.

The theory is nowadays discussed in multiple books including \cite{gb} and
\cite{iva}. We will follow these books along the way as we gradually unveil the
elegance and power of Gröbner bases in solving systems of polynomial equations.
Further information can be also found in \cite{igb} and \cite{gbs}.

\input{chapters/gb/algebraic_structures}

\input{chapters/gb/multivariate_polynomials}

\input{chapters/gb/monomial_orderings}

\input{chapters/gb/multivariate_division}